11 - The Large-Scale Geometry of Groups

Summary

What one really cares about are the inherent properties of the group, not the artefacts of a particular presentation.

–Martin Bridson

Changing Generators

There is a danger in working with a specific Cayley graph for a given group G. If you focus on a particular generating set, the results you get may not immediately translate into similar results when a different set of generators is used. Even worse, sometimes interesting properties hold in one Cayley graph but not in another, even though the group under consideration has not changed.

In this chapter we introduce some of the ways geometry can be imported into the study of infinite groups, which are independent of the choice of finite generating set. These are properties that always hold or always fail, no matter which finite set of generators one uses to construct a Cayley graph. Of course, such properties cannot focus too closely on a given Cayley graph, since changing generators changes the details, in other words the local structure, of the graphs. In Figure 11.1 we highlight this with two Cayley graphs for D₃. The location of the vertices has been kept constant in both pictures, which makes the second picture look a bit odd. When the generators change, so do the distances between vertices, the number of cycles, and so on.

Because changing generators can have dramatic consequences for the local structure of a Cayley graph, the properties we consider here are referred to as large-scale properties; many authors also use the term geometric properties.